PEDRO LEVIT KAUFMANN | Universidade Federal de São Paulo (UNIFESP) (original) (raw)

Papers by PEDRO LEVIT KAUFMANN

Esta dissertação tem por objetivo a apresentação de um estudo em espaços de Banach sobre os conju... more Esta dissertação tem por objetivo a apresentação de um estudo em espaços de Banach sobre os conjuntos nos quais determinados polinômios homogêneos contínuos são fracamente sequencialmente contínuos. Algumas propriedades desses conjuntos são estudadas e ilustradas com exemplos, em maior parte no espaço l p. Obtemos um fórmula para o conjunto de continuidade sequencial fraca do produto de dois polinômios e algumas consequências. Resultados mais fortes são obtidos quando restringimos nossos espaços de Banach a espaços com FDD incondicional e/ou separáveis. Os resultados estudados aqui foram obtidos por R. Aron e V. Dimant em [2].

We show that the set of Lebesgue integrable functions in [0,1] which are nowhere essentially boun... more We show that the set of Lebesgue integrable functions in [0,1] which are nowhere essentially bounded is spaceable, improving a result from [F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda. Lineability, spaceability, and algebrability of certain subsets of function spaces, Taiwanese J. Math., 13 (2009), no. 4, 1257--1269], and that it is strongly c-algebrable. We prove strong c-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesgue-nowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not 1-algebrable. We also show that there exists an infinite dimensional vector space S of differentiable functions such that each element of the C([0,1])-closure of S is a primitive to ...

We show that when C(K) does not have few operator -- in the sense of Koszmider [P. Koszmider, Ban... more We show that when C(K) does not have few operator -- in the sense of Koszmider [P. Koszmider, Banach spaces of continuous functions with few operators. Math. Ann. 300 (2004), no. 1, 151 - 183.] -- the sets of operators which are not weak multipliers is spaceable. This shows a contrast with what happens in general Banach spaces that do not have few operators. In addition, we show that there exist a C(K) space such that each operator on it is of the form gI+hJ+S, where g,h∈ C(K) and S is strictly singular, in connection to a result by Ferenczi [V. Ferenczi,Uniqueness of complex structure and real hereditarily indecomposable Banach spaces. Adv. Math. 213 (2007), no. 1, 462 - 488.].

We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q... more We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction f|_U is not in L^q(U). When (X,μ) satisfies some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p's but nowhere q-integrable for some other q's (0<p,q<∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question from Bernal-González, improve and complement recent spaceability and algebrability results from several authors and motivates new research directions in the field of spaceability.

We show that the set of Lebesgue integrable functions in [0, 1] which are nowhere essentially bou... more We show that the set of Lebesgue integrable functions in [0, 1] which are nowhere essentially bounded is spaceable, improving a result from [6], and that it is strongly c-algebrable. We prove strong c-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesguenowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not 1-algebrable. We also show that there exists an infinite dimensional vector space S of differentiable functions such that each element of the C([0, 1])-closure of S is a primitive to a Kurzweil integrable function, in connection to a classic spaceability result from [9].

We study combinatorial complexity of certain classes of products of intervals in R, from the poin... more We study combinatorial complexity of certain classes of products of intervals in R, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the VapnikChervonenkis dimension of the set of balls in l ∞ – which denotes R equipped with the sup norm – equals ⌊(3d+ 1)/2⌋.

It is known that for the one-dimensional case we obtain full generality (that is, (1) holds for e... more It is known that for the one-dimensional case we obtain full generality (that is, (1) holds for each differentiable F ) when we consider the equivalent Denjoy, Perron or Henstock-Kurzweil integrals (see e.g. [1]). The natural extension of the definition of the Henstock-Kurzweil integral for the two-dimensional case does not satisfy (2) even when U is an interval (that is, the cartesian product of compact intervals in the real line). The present work includes a brief discussion on previous attempts on modifying the Henstock-Kurzweil definition in order to obtain (2). The discussion is centralized on the gauge-based definitions introduced in [2], [3] and [4]. We then present some old (maybe revisited under a different perspective) and new results that can be summarized as follows: • The M1-integral introduced in [4] has the unpleasant property of being sensitive to rotations; this is shown using an example found in [5]. As a subproduct, we prove that the two-dimensional Henstock-Kurzw...

We say that a function f : (0,1) ! R is nowhere L q if, for each nonvoid open subset U of (0,1), ... more We say that a function f : (0,1) ! R is nowhere L q if, for each nonvoid open subset U of (0,1), the restriction f|U is not in L q (U). For a fixed 1 � p < 1, we will show that the set Sp .

We investigate the problem of classifying the Banach spaces mathrmLip0(C(K))\mathrm{Lip}_0(C(K))mathrmLip0(C(K)) for Hausdorff ... more We investigate the problem of classifying the Banach spaces mathrmLip0(C(K))\mathrm{Lip}_0(C(K))mathrmLip0(C(K)) for Hausdorff compacta KKK. In particular, sufficient conditions are established for a space mathrmLip0(C(K))\mathrm{Lip}_0(C(K))mathrmLip0(C(K)) to be isomorphic to mathrmLip0(c0(varGamma))\mathrm{Lip}_0(c_0(\varGamma))mathrmLip_0(c_0(varGamma)) for some uncountable set varGamma\varGammavarGamma.

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain... more We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group GGG equipped with an arbitrary compatible left-invariant metric ddd, the Lipschitz-free space over GGG, mathcalF(G,d)\mathcal{F}(G,d)mathcalF(G,d), satisfies the metric approximation property. We show also that, given a finitely generated group GGG, with its word metric ddd, from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, mathcalF(G,d)\mathcal{F}(G,d)mathcalF(G,d) has a Schauder basis. Examples and applications are discussed. In particular, for any net NNN in a real hyperbolic nnn-space mathbbHn\mathbb{H}^nmathbbHn, mathcalF(N)\mathcal{F}(N)mathcalF(N) has a Schauder basis.

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2016

Positioning, 2014

A new system's geo-referencing from space is entirely free from any GNSS (GPS or equivalent) syst... more A new system's geo-referencing from space is entirely free from any GNSS (GPS or equivalent) systems. The system addresses to various strategic and economic applications such as in remote clock synchronism, aircraft and balloon navigation, missile and smart bombs tracking, satellite orbital determination and remote target geo-positioning. The new geometry concept corresponds to an "inverted GPS" configuration, utilizing four ground-based reference stations, synchronized in time, installed at well known geodesic coordinates and a repeater in space, carried by an aircraft, balloon, satellite, etc. Signal transmitted by one of the reference bases is retransmitted by the transponder, received back by the four bases, producing four ranging measurements which are corrected for the time delays undergone in every retransmission. A minimization function was derived to compare the repeater's positions referred to at least two groups of three reference bases, to correct for the signal transit time at the repeater and propagation delays, and consequently to provide the accurate repeater position for each time interaction. Once the repeater's coordinates are known, the other determinations and applications become straightforward. The system solving algorithm and process performance has been demonstrated by simulations adopting a practical example with the transponder carried by an aircraft moving over bases and a target on the ground. Effects produced by reference clock synchronism uncertainties at the four bases on the measurements are reviewed.

ABSTRACT We present a two-dimensional nonabsolute gauge integral which satisfies several converge... more ABSTRACT We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theorems and a general divergence theorem, and at the same time admits a change of variables formula valid up to affine transformations - thus applicable to piecewise linear surfaces. Our approach is based on a modification of the \(M_1\)-integral presented in \cite{jks}, using triangle-based partitions.

ABSTRACT A brief description of a new concept for remote geopositioning time dissemination and na... more ABSTRACT A brief description of a new concept for remote geopositioning time dissemination and navigation applicable over regional areas currently being developed in Brazil is presented It aims at an independent Brazilian wide location system using concepts different from conventional GPS Galileo or GLONASS systems Its implementation becomes particularly important to allow independent comparisons validate measurements taken by other systems and assess the proposed system The new system utilizes at least three reference bases on the ground with precisely determined geodetic positions carrying synchronized clocks a transmitter in one central base master base emitting time coded marks and one repeater a satellite in the sky An algorithm and a digital telecommunication set up demonstrate its operation It is shown that in the master base it is necessary the knowledge of the satellite orbit with consistent precision Assuming a repeater put aboard an artificial satellite and a favorable geometry errors in the instantaneous determination of the position were studied Orbit propagation errors for one orbit or more are also presented One of the error sources which affects the orbit determination accuracy comes from the measurement system involving the ground segment geometry between the orbit and the bases and equipment For the sake of this work three measurement error sources were considered here the station location error the oscillator stability and miscellaneous additional sources

The objective of this work is to present a new algebraic solution for the problem of remote deter... more The objective of this work is to present a new algebraic solution for the problem of remote determination of geographic coordinates of a target, using a new remote geopositioning system being developed in Brazil. It can be useful for double-check measurements obtained with other methods, for certain critical applications, being capable to perform independently from them. This system requires three-reference bases on the surface of the earth with synchronized clocks and a repeater in space. Calculations are derived from measurements of propagation time of clock signal transmitted by one base to all bases and target, via a transponder in space. The algorithm also provides the "instantaneous" determination of the repeater position in space and brings other applications in navigation and remote clock synchronization. The algorithm has been successfully tested through computational software.

Esta dissertação tem por objetivo a apresentação de um estudo em espaços de Banach sobre os conju... more Esta dissertação tem por objetivo a apresentação de um estudo em espaços de Banach sobre os conjuntos nos quais determinados polinômios homogêneos contínuos são fracamente sequencialmente contínuos. Algumas propriedades desses conjuntos são estudadas e ilustradas com exemplos, em maior parte no espaço l p. Obtemos um fórmula para o conjunto de continuidade sequencial fraca do produto de dois polinômios e algumas consequências. Resultados mais fortes são obtidos quando restringimos nossos espaços de Banach a espaços com FDD incondicional e/ou separáveis. Os resultados estudados aqui foram obtidos por R. Aron e V. Dimant em [2].

We show that the set of Lebesgue integrable functions in [0,1] which are nowhere essentially boun... more We show that the set of Lebesgue integrable functions in [0,1] which are nowhere essentially bounded is spaceable, improving a result from [F. J. García-Pacheco, M. Martín, and J. B. Seoane-Sepúlveda. Lineability, spaceability, and algebrability of certain subsets of function spaces, Taiwanese J. Math., 13 (2009), no. 4, 1257--1269], and that it is strongly c-algebrable. We prove strong c-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesgue-nowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not 1-algebrable. We also show that there exists an infinite dimensional vector space S of differentiable functions such that each element of the C([0,1])-closure of S is a primitive to ...

We show that when C(K) does not have few operator -- in the sense of Koszmider [P. Koszmider, Ban... more We show that when C(K) does not have few operator -- in the sense of Koszmider [P. Koszmider, Banach spaces of continuous functions with few operators. Math. Ann. 300 (2004), no. 1, 151 - 183.] -- the sets of operators which are not weak multipliers is spaceable. This shows a contrast with what happens in general Banach spaces that do not have few operators. In addition, we show that there exist a C(K) space such that each operator on it is of the form gI+hJ+S, where g,h∈ C(K) and S is strictly singular, in connection to a result by Ferenczi [V. Ferenczi,Uniqueness of complex structure and real hereditarily indecomposable Banach spaces. Adv. Math. 213 (2007), no. 1, 462 - 488.].

We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q... more We say that a real-valued function f defined on a positive Borel measure space (X,μ) is nowhere q-integrable if, for each nonvoid open subset U of X, the restriction f|_U is not in L^q(U). When (X,μ) satisfies some natural properties, we show that certain sets of functions defined in X which are p-integrable for some p's but nowhere q-integrable for some other q's (0<p,q<∞) admit a variety of large linear and algebraic structures within them. The presented results answer a question from Bernal-González, improve and complement recent spaceability and algebrability results from several authors and motivates new research directions in the field of spaceability.

We show that the set of Lebesgue integrable functions in [0, 1] which are nowhere essentially bou... more We show that the set of Lebesgue integrable functions in [0, 1] which are nowhere essentially bounded is spaceable, improving a result from [6], and that it is strongly c-algebrable. We prove strong c-algebrability and non-separable spaceability of the set of functions of bounded variation which have a dense set of jump discontinuities. Applications to sets of Lebesguenowhere-Riemann integrable and Riemann-nowhere-Newton integrable functions are presented as corollaries. In addition we prove that the set of Kurzweil integrable functions which are not Lebesgue integrable is spaceable (in the Alexievicz norm) but not 1-algebrable. We also show that there exists an infinite dimensional vector space S of differentiable functions such that each element of the C([0, 1])-closure of S is a primitive to a Kurzweil integrable function, in connection to a classic spaceability result from [9].

We study combinatorial complexity of certain classes of products of intervals in R, from the poin... more We study combinatorial complexity of certain classes of products of intervals in R, from the point of view of Vapnik-Chervonenkis geometry. As a consequence of the obtained results, we conclude that the VapnikChervonenkis dimension of the set of balls in l ∞ – which denotes R equipped with the sup norm – equals ⌊(3d+ 1)/2⌋.

It is known that for the one-dimensional case we obtain full generality (that is, (1) holds for e... more It is known that for the one-dimensional case we obtain full generality (that is, (1) holds for each differentiable F ) when we consider the equivalent Denjoy, Perron or Henstock-Kurzweil integrals (see e.g. [1]). The natural extension of the definition of the Henstock-Kurzweil integral for the two-dimensional case does not satisfy (2) even when U is an interval (that is, the cartesian product of compact intervals in the real line). The present work includes a brief discussion on previous attempts on modifying the Henstock-Kurzweil definition in order to obtain (2). The discussion is centralized on the gauge-based definitions introduced in [2], [3] and [4]. We then present some old (maybe revisited under a different perspective) and new results that can be summarized as follows: • The M1-integral introduced in [4] has the unpleasant property of being sensitive to rotations; this is shown using an example found in [5]. As a subproduct, we prove that the two-dimensional Henstock-Kurzw...

We say that a function f : (0,1) ! R is nowhere L q if, for each nonvoid open subset U of (0,1), ... more We say that a function f : (0,1) ! R is nowhere L q if, for each nonvoid open subset U of (0,1), the restriction f|U is not in L q (U). For a fixed 1 � p < 1, we will show that the set Sp .

We investigate the problem of classifying the Banach spaces mathrmLip0(C(K))\mathrm{Lip}_0(C(K))mathrmLip0(C(K)) for Hausdorff ... more We investigate the problem of classifying the Banach spaces mathrmLip0(C(K))\mathrm{Lip}_0(C(K))mathrmLip0(C(K)) for Hausdorff compacta KKK. In particular, sufficient conditions are established for a space mathrmLip0(C(K))\mathrm{Lip}_0(C(K))mathrmLip0(C(K)) to be isomorphic to mathrmLip0(c0(varGamma))\mathrm{Lip}_0(c_0(\varGamma))mathrmLip_0(c_0(varGamma)) for some uncountable set varGamma\varGammavarGamma.

We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain... more We study Lipschitz-free spaces over compact and uniformly discrete metric spaces enjoying certain high regularity properties - having group structure with left-invariant metric. Using methods of harmonic analysis we show that, given a compact metrizable group GGG equipped with an arbitrary compatible left-invariant metric ddd, the Lipschitz-free space over GGG, mathcalF(G,d)\mathcal{F}(G,d)mathcalF(G,d), satisfies the metric approximation property. We show also that, given a finitely generated group GGG, with its word metric ddd, from a class of groups admitting a certain special type of combing, which includes all hyperbolic groups and Artin groups of large type, mathcalF(G,d)\mathcal{F}(G,d)mathcalF(G,d) has a Schauder basis. Examples and applications are discussed. In particular, for any net NNN in a real hyperbolic nnn-space mathbbHn\mathbb{H}^nmathbbHn, mathcalF(N)\mathcal{F}(N)mathcalF(N) has a Schauder basis.

Bulletin of the Belgian Mathematical Society - Simon Stevin, 2016

Positioning, 2014

A new system's geo-referencing from space is entirely free from any GNSS (GPS or equivalent) syst... more A new system's geo-referencing from space is entirely free from any GNSS (GPS or equivalent) systems. The system addresses to various strategic and economic applications such as in remote clock synchronism, aircraft and balloon navigation, missile and smart bombs tracking, satellite orbital determination and remote target geo-positioning. The new geometry concept corresponds to an "inverted GPS" configuration, utilizing four ground-based reference stations, synchronized in time, installed at well known geodesic coordinates and a repeater in space, carried by an aircraft, balloon, satellite, etc. Signal transmitted by one of the reference bases is retransmitted by the transponder, received back by the four bases, producing four ranging measurements which are corrected for the time delays undergone in every retransmission. A minimization function was derived to compare the repeater's positions referred to at least two groups of three reference bases, to correct for the signal transit time at the repeater and propagation delays, and consequently to provide the accurate repeater position for each time interaction. Once the repeater's coordinates are known, the other determinations and applications become straightforward. The system solving algorithm and process performance has been demonstrated by simulations adopting a practical example with the transponder carried by an aircraft moving over bases and a target on the ground. Effects produced by reference clock synchronism uncertainties at the four bases on the measurements are reviewed.

ABSTRACT We present a two-dimensional nonabsolute gauge integral which satisfies several converge... more ABSTRACT We present a two-dimensional nonabsolute gauge integral which satisfies several convergence theorems and a general divergence theorem, and at the same time admits a change of variables formula valid up to affine transformations - thus applicable to piecewise linear surfaces. Our approach is based on a modification of the \(M_1\)-integral presented in \cite{jks}, using triangle-based partitions.

ABSTRACT A brief description of a new concept for remote geopositioning time dissemination and na... more ABSTRACT A brief description of a new concept for remote geopositioning time dissemination and navigation applicable over regional areas currently being developed in Brazil is presented It aims at an independent Brazilian wide location system using concepts different from conventional GPS Galileo or GLONASS systems Its implementation becomes particularly important to allow independent comparisons validate measurements taken by other systems and assess the proposed system The new system utilizes at least three reference bases on the ground with precisely determined geodetic positions carrying synchronized clocks a transmitter in one central base master base emitting time coded marks and one repeater a satellite in the sky An algorithm and a digital telecommunication set up demonstrate its operation It is shown that in the master base it is necessary the knowledge of the satellite orbit with consistent precision Assuming a repeater put aboard an artificial satellite and a favorable geometry errors in the instantaneous determination of the position were studied Orbit propagation errors for one orbit or more are also presented One of the error sources which affects the orbit determination accuracy comes from the measurement system involving the ground segment geometry between the orbit and the bases and equipment For the sake of this work three measurement error sources were considered here the station location error the oscillator stability and miscellaneous additional sources

The objective of this work is to present a new algebraic solution for the problem of remote deter... more The objective of this work is to present a new algebraic solution for the problem of remote determination of geographic coordinates of a target, using a new remote geopositioning system being developed in Brazil. It can be useful for double-check measurements obtained with other methods, for certain critical applications, being capable to perform independently from them. This system requires three-reference bases on the surface of the earth with synchronized clocks and a repeater in space. Calculations are derived from measurements of propagation time of clock signal transmitted by one base to all bases and target, via a transponder in space. The algorithm also provides the "instantaneous" determination of the repeater position in space and brings other applications in navigation and remote clock synchronization. The algorithm has been successfully tested through computational software.